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Lemma 6.7. Let \( Q\left( {{z}_{1},{z}_{2},{z}_{3},{z}_{4}}\right) \) be a quadrilateral with module 1, and let \( {s}_{1} \) and \( {s}_{2} \) denote the euclidean distances in \( Q \) between the sides \( \left( {{z}_{1},{z}_{2}}\right) ,\left( {{z}_{3},{z}_{4}}\right) \) and \( \left( {{z}_{2},{z}_{3}}\right) ,\left... | Proof. We may assume that among the arcs which join the sides \( \left( {{z}_{2},{z}_{3}}\right) \) and \( \left( {{z}_{4},{z}_{1}}\right) \) in \( Q \) there is a \( {\gamma }_{0} \) of length \( {s}_{2} \) . Let \( {z}_{0} \) be the point which divides \( {\gamma }_{0} \) into two parts of length \( {s}_{2}/2 \) . Se... | Yes |
Theorem 6.6. A Jordan domain whose boundary satisfies the arc condition is a quasidisc. | Proof. Let \( C \) be a Jordan curve which satisfies the arc condition and bounds the domains \( {A}_{1} \) and \( {A}_{2} \) . Choose four points \( {z}_{1},{z}_{2},{z}_{3},{z}_{4} \) on \( C \) such that \( {A}_{1}\left( {{z}_{1},{z}_{2},{z}_{3},{z}_{4}}\right) \) is a quadrilateral with module 1 . We shall derive an... | Yes |
Theorem 6.7. Let \( C \) be a \( K \) -quasicircle passing through \( \infty \), and \( {z}_{1},{z}_{2},{z}_{3} \) finite points of \( C \) such that \( {z}_{2} \) lies between \( {z}_{1} \) and \( {z}_{3} \) . Then\n\n\[ \left| {{z}_{1} - {z}_{2}}\right| + \left| {{z}_{2} - {z}_{3}}\right| \leq c\left( K\right) \left|... | Proof. Let \( f \) be a \( K \) -quasiconformal mapping of the plane which maps the real axis onto \( C \) such that \( f\left( \infty \right) = \infty \) . Denote \( {x}_{i} = {f}^{-1}\left( {z}_{i}\right), i = 1,2,3 \), and \( {C}_{1} = \left\{ {w\left| \right| w - {x}_{1}\left| = \right| {x}_{1} - {x}_{2} \mid }\rig... | Yes |
Theorem 1.2 (Area Theorem). Let \( f \) be a univalent meromorphic function in the domain \( \{ z\left| \right| z \mid > 1\} \), with a power series expansion\n\n\[ f\left( z\right) = z + \mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n}{z}^{-n}. \]\n\nThen\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }n{\left| {b}_{n}... | Proof. Let \( {C}_{\rho } \) be the image of the circle \( \left| z\right| = \rho > 1 \) under \( f \) . The finite domain bounded by \( {C}_{\rho } \) has the area\n\n\[ {m}_{\rho } = \frac{i}{2}{\int }_{{C}_{\rho }}{wd}\bar{w} \]\n\nSubstituting \( w = f\left( z\right) \) and considering (1.14) we obtain\n\n\[ {m}_{\... | Yes |
Theorem 1.3. Iff is a conformal mapping of a disc, then\n\n\[ \n\\begin{Vmatrix}{S}_{f}\\end{Vmatrix} \\leq 6\\text{.}\n\]\n\n(1.20)\n\nThe bound is sharp. | Proof. By formula (1.12) it does not matter in which disc \( f \) is defined. We suppose that \( f \) is a conformal mapping of the unit disc \( D \) . Let us choose a point \( {z}_{0} \\in D \) and estimate \( \\left| {{S}_{f}\\left( {z}_{0}\\right) }\\right| \\eta {\\left( {z}_{0}\\right) }^{-2} = {\\left( 1 - {\\lef... | Yes |
Theorem 2.1. If \( A \) is Möbius equivalent to a convex domain, then\n\n\[ \delta \left( A\right) \leq 2\text{.} \]\n\nEquality holds if \( A \) is the image of a parallel strip under a Möbius transformation. | Proof. We may assume that \( A \) itself is convex. Let \( f \) be an arbitrary conformal mapping of \( D \) onto \( A \) . In view of (2.2), inequality (2.9) follows if we prove that \( \left| {{S}_{f}\left( 0\right) }\right| \leq 2 \) . Since we may replace \( f \) by the function \( z \rightarrow {cf}\left( {z{e}^{i... | Yes |
Theorem 2.2. Let \( A \) be Möbius equivalent to a domain with boundary rotation \( \leq {k\pi } \) . If \( k \leq 4 \), then \[ \delta \left( A\right) \leq \frac{{2k} + 4}{6 - k}. \] | The main lines of the proof are the same as those in Theorem 2.1. After similar initial remarks we start from (2.10), assuming this time that \( {S}_{f}\left( 0\right) < 0 \) . In the first line of (2.11) we now ignore the third integral and conclude that \[ \left| {{S}_{f}\left( 0\right) }\right| \leq \frac{1}{2}{\lef... | No |
Theorem 2.3. Let \( A \) and \( {A}^{\prime } \) be domains conformally equivalent to a disc and \( f : A \rightarrow {A}^{\prime } \) a conformal mapping. Then\n\n\[ \n{\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} \leq \delta \left( A\right) + \delta \left( {A}^{\prime }\right) \n\]\n\n(2.13)\n\nThe estimate is sharp for ... | Proof. Let \( h \) be a conformal mapping of the unit disc \( D \) onto \( A \) . From \( f = \left( {f \circ h}\right) \circ {h}^{-1} \) we conclude that \( {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} = {\begin{Vmatrix}{S}_{f \circ h} - {S}_{h}\end{Vmatrix}}_{D} \) . Since\n\n\[ \n{\begin{Vmatrix}{S}_{h}\end{Vmatrix}}_{... | Yes |
Theorem 2.4. Let \( A \) be a domain conformally equivalent to a disc. Then\n\n\[ \n{\sigma }_{0}\left( A\right) = \delta \left( A\right) + 6 \n\] | Proof. We write the definition of \( {\sigma }_{0}\left( A\right) \) in the form\n\n\[ \n{\sigma }_{0}\left( A\right) = \mathop{\sup }\limits_{{A}^{\prime }}\left\{ {{\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} \mid f : A \rightarrow {A}^{\prime }\text{ conformal }}\right\} .\n\]\n\nThen it follows from Theorem 2.3 that\n... | Yes |
Let \( \mu \) be a measurable function in the plane with bounded support and \( \parallel \mu {\parallel }_{\infty } < 1 \) . Let \( z \rightarrow f\left( {z, w}\right) \) be the quasiconformal mapping of the plane with complex dilatation \( {w\mu } \) and with the property \( \lim \left( {f\left( {z, w}\right) - z}\ri... | Proof. By Theorem I.4.3,\n\n\[ f\left( z\right) = f\left( {z,1}\right) = z + \mathop{\sum }\limits_{{i = 1}}^{\infty }T{\varphi }_{i}\left( \mu \right) \left( z\right) ,\]\n\nwhere we now write \( {\varphi }_{i}\left( \mu \right) \) instead of \( {\varphi }_{i} \) to accentuate the dependence of \( {\varphi }_{i} \) on... | Yes |
Let \( \mu \) be a measurable function in the plane which vanishes in the upper half-plane and for which \( \parallel \mu {\parallel }_{\infty } < 1 \) . Let \( {f}_{w\mu } \) be the quasiconformal mapping of the plane with complex dilatation \( {w\mu } \) which keeps the points \( 0,1,\infty \) fixed. Then the functio... | Let \( g \) be the Möbius transformation which maps the points \( 0,1,\infty \) on the points \( - 1, i,1 \), respectively, and \( {\widetilde{f}}_{wv} \) a quasiconformal mapping of the plane whose complex dilatation \( {wv} \) agrees with that of \( {f}_{w\mu } \circ {g}^{-1} \) . Then\n\n\[ \mu \left( z\right) = v\l... | Yes |
Theorem 3.2. Let \( f \) be a quasiconformal mapping of the plane which has the complex dilatation \( \mu \) and which is conformal in a simply connected domain \( A \) with at least two boundary points. Then\n\n\[ \n{\begin{Vmatrix}{S}_{f \mid A}\end{Vmatrix}}_{A} \leq {\sigma }_{0}\left( A\right) \parallel \mu {\para... | Proof. If \( g \) is a Möbius transformation, we can replace \( f \) by \( f \circ g \) without changing the norms of either the Schwarzian derivative or the complex dilatation. Also, \( {\sigma }_{0}\left( A\right) = {\sigma }_{0}\left( {{g}^{-1}\left( A\right) }\right) \) . We may therefore assume that \( \infty \in ... | Yes |
Theorem 3.3. If \( A \) is a \( K \) -quasidisc, then\n\n\[ \delta \left( A\right) \leq 6\frac{{K}^{2} - 1}{{K}^{2} + 1} \] | Proof. By Lemma I.6.2, the domain \( A \) is the image of the upper half-plane \( H \) under a \( {K}^{2} \) -quasiconformal mapping \( f \) of the plane which is conformal in \( H \) . By Theorem 3.2,\n\n\[ {\begin{Vmatrix}{S}_{f \mid H}\end{Vmatrix}}_{H} \leq 6\frac{{K}^{2} - 1}{{K}^{2} + 1}. \]\n\nOn the other hand,... | Yes |
Theorem 3.4. Let \( f \in {\sum }_{k} \) and \( k < {k}_{0} < 1 \) . As \( k \rightarrow 0 \) ,\n\n\[ f\left( z\right) = z - \frac{1}{\pi }{\iint }_{D}\frac{\mu \left( \zeta \right) }{\zeta - z}{d\xi d\eta } + O\left( {k}^{2}\right) \]\n\nin the whole plane. Here \( \left| {O\left( {k}^{2}\right) }\right| \leq c{k}^{2}... | Proof. If \( p > 2 \) and \( {k}_{0}\parallel H{\parallel }_{p} < 1 \), we see from formula (4.15) in I.4.4 that\n\n\[ \mathop{\sum }\limits_{{i = 2}}^{\infty }\left| {T{\varphi }_{i}\left( z\right) }\right| \leq {c}_{p}^{\prime }\mathop{\sum }\limits_{{i = 2}}^{\infty }{\left( k\parallel H{\parallel }_{p}\right) }^{i}... | No |
Corollary 3.2. The functions \( f \in {\sum }_{k} \) satisfy the asymptotic inequality\n\n\[ \left| {f\left( z\right) - z}\right| \leq \frac{k}{\pi }{\iint }_{D}\frac{d\xi d\eta }{\left| \zeta - z\right| } + c{k}^{2}. \] | If\n\n\[ \mu \left( \zeta \right) = k{e}^{i\theta }\frac{\zeta - z}{\left| \zeta - z\right| }\;\text{ a.e.,} \]\n\nthen\n\n\[ \left| {f\left( z\right) - z}\right| = \frac{k}{\pi }{\iint }_{D}\frac{d\xi d\eta }{\left| \zeta - z\right| } + O\left( {k}^{2}\right) . \] | Yes |
Theorem 3.5. In the class \( {\sum }_{k} \), \[ \left| {b}_{n}\right| \leq \frac{2k}{n + 1} + c{k}^{2},\;n = 1,2,\ldots , \] with \( c \leq {n}^{-1/2}{\left( 1 - k\right) }^{-1} \). If \[ {f}_{n}\left( z\right) = \left\{ \begin{array}{ll} {\left( {z}^{\left( {n + 1}\right) /2} + k{z}^{-\left( {n + 1}\right) /2}\right) ... | Proof. We have \[ T{\varphi }_{i}\left( z\right) = - \frac{1}{\pi }{\iint }_{D}\frac{{\varphi }_{i}\left( \zeta \right) }{\zeta - z}{d\xi d\eta } = \frac{1}{\pi }\mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {{\iint }_{D}{\varphi }_{i}\left( \zeta \right) {\zeta }^{n - 1}{d\xi d\eta }}\right) {z}^{-n} \] for \( \left... | Yes |
Theorem 3.6. Let \( \Phi \) be an analytic functional on \( \sum \) which vanishes for the identity mapping. Then \( M\left( k\right) /k \) is non-decreasing on the interval \( \left( {0,1}\right) \) . | Proof. Fix \( k \) and \( {k}^{\prime },0 < k < {k}^{\prime } < 1 \), and choose an arbitrary mapping \( {f}_{0} \in {\sum }_{k} \) . Let \( \mu \) be the complex dilatation of some extension of \( {f}_{0},\parallel \mu {\parallel }_{\infty } \leq k \) . Consider the mappings \( f \in {\sum }_{{k}^{\prime }} \) which h... | Yes |
Corollary 3.3 (Majorant Principle). If \( \Phi \) is an analytic functional on \( \sum \) which vanishes for the identity mapping, then\n\n\[ \mathop{\max }\limits_{{f \in {\sum }_{k}}}\left| {\Phi \left( f\right) }\right| \leq k\mathop{\max }\limits_{{f \in \sum }}\left| {\Phi \left( f\right) }\right| \]\n\n(3.13)\n\n... | Proof. Inequality (3.13) follows immediately from Theorem 3.6 if we let \( k \rightarrow 1 \) .\n\nSuppose that equality holds in (3.13) for some value \( k,0 < k < 1 \) . Let \( {f}_{k} \) be extremal in this \( {\sum }_{k} \) and \( \mu \) the complex dilatation of its extension. For functions \( f \) with complex di... | Yes |
Theorem 3.7. In the class \( {\sum }_{k} \) , \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }n{\left| {b}_{n}\right| }^{2} \leq {k}^{2} \] (3.16) The estimate is sharp. | Proof. Given an arbitrary function \( f \in {\sum }_{k} \) with the coefficients \( {b}_{n} \), we set \( {\lambda }_{n} = {\left| {b}_{n}\right| }^{2}/{b}_{n}^{2} \) if \( {b}_{n} \neq 0 \) ; otherwise \( {\lambda }_{n} = 1 \) . Let \( \mu \) be the complex dilatation of the extended \( f \), and \( {b}_{n}\left( w\ri... | Yes |
Theorem 4.2. A Schwarzian domain is a quasidisc. | Proof. Let \( A \) be an \( a \) -Schwarzian domain. Then \( A \) is trivially \( {a}^{\prime } \) -Schwarzian for \( {a}^{\prime } \leq a \) . We may suppose, therefore, that \( a \leq 2 \) . (In III. 5 we shall show that, in fact, no domain \( A \) is \( a \) -Schwarzian for \( a > 2 \), but here this result is not n... | Yes |
Theorem 5.2. Let \( f \) be meromorphic in a disc. If\n\n\[ \begin{Vmatrix}{S}_{f}\end{Vmatrix} \leq 2 \]\n\nthen \( f \) is univalent. The bound 2 is best possible. | Proof. Consider functions \( {f}_{n}, n = 1,2,\ldots \), which are meromorphic in the given disc, fix three points of the disc, and have Schwarzians \( \left( {1 - 1/n}\right) {S}_{f} \) ; by Theorem 1.1 such functions exist. Since \( \begin{Vmatrix}{S}_{{f}_{n}}\end{Vmatrix} < 2 \), every \( {f}_{n} \) is univalent ow... | Yes |
Theorem 5.4. Let \( f \) be meromorphic and satisfy\n\n\[ \n\\left| {{S}_{f}\\left( z\\right) }\\right| < \\frac{2}{{\\left( 1 - {\\left| z\\right| }^{2}\\right) }^{2}}\n\]\n\nin the unit disc. Then \( f \) is univalent and has a homeomorphic extension to the plane. | Proof. By Theorem 5.2, \( f \) is univalent. The image \( f\\left( D\\right) \) is a Jordan domain if and only if \( f \) has a homeomorphic extension to the plane. Hence, if a homeomorphic extension does not exist, then by Theorem \( {5.3}, f\\left( D\\right) \) is the image of the parallel strip \( A \) under a Möbiu... | Yes |
Theorem 1.1. Every point of the universal Teichmüller space can be represented by a real analytic quasiconformal mapping \( f \in F \) or by a real analytic complex dilatation \( \mu \in B \) . | Proof. The result follows immediately from Theorem I.5.3. (For a complete proof, see II.5.2.) | No |
Theorem 1.2. The complex dilatations \( \mu \) and \( v \) are equivalent if and only if the conformal mappings \( {f}_{\mu }\left| {H}^{\prime }\right| \) and \( {f}_{v} \mid {H}^{\prime } \) coincide. | Proof. Suppose first that \( {f}_{\mu }\left| {{H}^{\prime } = {f}_{v}}\right| {H}^{\prime } \) . The mappings \( {f}_{\mu } \circ {\left( {f}^{\mu }\right) }^{-1} \) and \( {f}_{v} \circ {\left( {f}^{v}\right) }^{-1} \) are both conformal in the upper half-plane \( H \), which they map onto the same quasidisc. Because... | Yes |
Lemma 1.1. Let \( h \) be a normalized quasisymmetric function. Then the sewing problem has a unique normalized pair of solutions. | Proof. Given a function \( h \in X \), there is a mapping \( {f}^{\mu } \in F \) such that \( {f}^{\mu } \mid \mathbb{R} = h \) . Then\n\n\[ \n{f}_{1} = \left( {{f}_{\mu } \mid H}\right) \circ {\left( {f}^{\mu }\right) }^{-1},\;{f}_{2} = {f}_{\mu } \mid {H}^{\prime }, \]\n\n\nis a solution of the sewing problem. This c... | Yes |
Theorem 1.3. Two points \( \left\lbrack {f}^{\mu }\right\rbrack ,\left\lbrack {f}^{v}\right\rbrack \in T \) are inverse elements of the group \( T \) if and only if the quasidiscs \( {f}_{\mu }\left( H\right) \) and \( {f}_{v}\left( {H}^{\prime }\right) \) are mirror images with respect to the real axis. | Proof. Assume first that \( \left\lbrack {f}^{\mu }\right\rbrack \) and \( \left\lbrack {f}^{v}\right\rbrack \) are inverse; we can then take \( {f}^{v} = {\left( {f}^{\mu }\right) }^{-1} \) . Let \( {f}_{{\mu }^{ * }} \) be the quasiconformal mapping of the plane which fixes the points \( 0,1,\infty \) and whose compl... | Yes |
Lemma 2.1. The functions \( \tau ,{\tau }_{1} \) and \( {\tau }_{2} \) are the same. | Proof. Clearly, \( \tau \leq {\tau }_{1} \) . If \( w \in W \), then \( g = w \circ {f}_{0} \in q \), so that \( {\tau }_{1} \leq {\tau }_{2} \) . Finally, if \( f \in p, g \in q \), then \( g \circ {f}^{-1} \in W \), and so \( {\tau }_{2} \leq \tau \) . | No |
Theorem 2.1. The universal Teichmüller space is pathwise connected. | Proof. Consider the origin of \( T \), i.e., the point represented by the function of \( B \) which is identically zero, and an arbitrary point \( p \in T \) represented by \( \mu \) . For \( 0 \leq t \leq 1 \), let \( {p}_{t} \) be the point represented by the function \( {t\mu } \) of \( B \) . Then\n\n\[ \beta \left... | Yes |
Theorem 2.2. If \( \mu \) is an extremal complex dilatation for the point \( p \in T \), then\n\n\[ \n{\mu }_{t} = \frac{{\left( 1 + \left| \mu \right| \right) }^{t} - {\left( 1 - \left| \mu \right| \right) }^{t}}{{\left( 1 + \left| \mu \right| \right) }^{t} + {\left( 1 - \left| \mu \right| \right) }^{t}}\frac{\mu }{\l... | Proof. From (2.5) we see that \( {\mu }_{t}\left( z\right) \) is the point which divides the hyperbolic length (in the unit disc) of the line segment from 0 to \( \mu \left( z\right) \) in the ratio \( t : \left( {1 - t}\right) \) (cf. formula (4.16) in I.4.7).\n\nIf \( {f}^{\mu } \) has maximal dilatation \( K \), the... | Yes |
Theorem 2.3. The universal Teichmüller space is complete. | Proof. In view of statement \( {2}^{ \circ } \) in Lemma 2.2, it is enough to observe that if a Cauchy sequence contains a convergent subsequence, then the sequence itself is convergent. | No |
Theorem 3.1. The group isomorphism\n\n\\[ \n\\left\\lbrack f\\right\\rbrack \\rightarrow f \\mid \\mathbb{R} \n\\]\n\n(3.1)\n\nis a homeomorphism of \\( \\left( {T,\\tau }\\right) \\) onto \\( \\left( {X,\\rho }\\right) \\) . | Proof. We proved in 1.1 that (3.1) is a bijection of \\( T \\) onto \\( X \\) . From (2.4) and the left-hand inequality (5.10) in I.5.7 it follows that\n\n\\[ \n\\rho \\left( {{f}_{1}\\left| {\\mathbb{R},{f}_{2}}\\right| \\mathbb{R}}\\right) \\leq \\tau \\left( {\\left\\lbrack {f}_{1}\\right\\rbrack ,\\left\\lbrack {f}... | Yes |
Theorem 3.2. The universal Teichmüller space is contractible. | Proof. Every point of \( T \) is an equivalence class \( \left\lbrack {s\left( h\right) }\right\rbrack, h \in X \) . We show that\n\n\[ \left( {\left\lbrack {s\left( h\right) }\right\rbrack, t}\right) \rightarrow \left\lbrack {\left( {1 - t}\right) s\left( h\right) }\right\rbrack \]\n\n(3.3)\n\ndeforms \( T \) continuo... | Yes |
Theorem 3.3. The universal Teichmüller space is not a topological group. | Proof. The theorem follows if we find an \( \left\lbrack f\right\rbrack \in T \) and a sequence of points \( \left\lbrack {g}_{n}\right\rbrack \in T \), such that \( \left\lbrack {g}_{n}\right\rbrack \) tends to \( \left\lbrack g\right\rbrack \) but \( \left\lbrack {f \circ {g}_{n}}\right\rbrack \) does not tend to \( ... | Yes |
Theorem 4.1. The mapping\n\n\\[ \n\\left\\lbrack \\mu \\right\\rbrack \\rightarrow {S}_{{f}_{\\mu } \\mid H} \n\\]\n\n(4.7)\n\nis a homeomorphism of the universal Teichmüller space onto its image in \\( Q \\) . | Proof. We noted already in 4.1 that (4.7) is well defined in \\( T \\) . If \\( \\left\\lbrack \\mu \\right\\rbrack \\) and \\( \\left\\lbrack v\\right\\rbrack \\) have the same image, it follows from the normalization that \\( {f}_{\\mu }\\left| {H = {f}_{v}}\\right| H \\), i.e., \\( \\mu \\) and \\( v \\) are equival... | Yes |
Theorem 4.2. The set \( T\left( 1\right) \) is the interior of \( U \) . | Proof. We prove first that \( T\left( 1\right) \) is an open subset of \( Q \) . Fix an arbitrary point \( {S}_{f} \) of \( T\left( 1\right) \) . For \( {S}_{h} \in Q \) we write \( g = h \circ {f}^{-1} \), and conclude that \( g \) is meromorphic in the quasidisc \( f\left( H\right) \) . By Theorem II.4.1, there exist... | Yes |
Theorem 4.3. The closure of \( T\left( 1\right) \) is a proper subset of \( U \) . | Proof. Let \( G \) be the domain defined above and \( \varepsilon > 0 \) the associated constant. If \( h \) is a conformal mapping of the upper half-plane onto \( G \), we prove that \( {S}_{h} \) does not lie in the closure of \( T\left( 1\right) \) .\n\nConsider an arbitrary point \( {S}_{w} \) of the neighborhood \... | Yes |
Lemma 5.2. Let \( A \) be a quasidisc which is contained in a domain \( {B}_{k} \) Möbius equivalent to the sector \( {A}_{k} = \{ z \mid \left| {\arg z}\right| < {k\pi }/2\} \) . If \( 0 < k \leq 1 \), assume that a vertex \( v \) of \( {B}_{k} \) lies on \( \partial A \) . Then\n\n\[ \n{\sigma }_{I}\left( A\right) \l... | Proof. Suppose first that \( 0 < k \leq 1 \) . Let \( g \) be a Möbius transformation mapping \( {B}_{k} \) onto \( {A}_{k} \) with \( g\left( v\right) = 0 \) . Set \( f\left( z\right) = \log g\left( z\right) \) . Then \( f\left( A\right) \) is not a quasidisc, and so by (5.5), \( {\sigma }_{I}\left( A\right) \leq {\be... | Yes |
Lemma 5.3. Let \( A \) be a quasidisc. If every two-point subset of \( A \) is contained in the closure of a quasidisc \( B \subset A \) for which \( {\sigma }_{I}\left( B\right) \geq m \), then\n\n\[ \n{\sigma }_{I}\left( A\right) \geq m\text{.} \n\] | Proof. Let an \( \varepsilon > 0 \) be given. There exists a meromorphic function \( f \) in \( A \) for which \( {\begin{Vmatrix}{S}_{f}\end{Vmatrix}}_{A} < {\sigma }_{I}\left( A\right) + \varepsilon \) but which is not univalent. Let \( {z}_{1} \) and \( {z}_{2} \) be two different points of \( A \) such that \( f\le... | Yes |
Theorem 5.3. For all domains A conformally equivalent to a disc,\n\n\[ \n{\sigma }_{I}\left( A\right) \leq 2\text{.} \n\]\n\nEquality holds if and only if \( A \) is a disc. | Proof. Let \( A \) be an arbitrary quasidisc. Every Jordan domain is Möbius equivalent to a subdomain of \( H \) having 0 and \( \infty \) as boundary points. We may assume, therefore, that \( A \) itself is such a domain.\n\nIn \( A \), we consider the function \( z \rightarrow f\left( z\right) = \log z \), for which ... | Yes |
Theorem 1.1. Every orientable \( {C}^{2} \) -surface in \( {\mathbb{R}}^{3} \) can be made into a Riemann surface. | Proof. Let \( S \) be an orientable \( {C}^{2} \) -surface. Consider an arbitrary local parameter of \( S \) inducing local coordinates \( z \) in a domain \( A \) of the complex plane. The theorem follows if we can transform the \( z \) -coordinates diffeomorphically so that the new coordinates are isothermal.\n\nExpr... | No |
Theorem 2.1 (Monodromy Theorem). Let \( \left( {W, f}\right) \) be an unlimited covering surface of a surface \( S \), and \( {\gamma }_{0} \) and \( {\gamma }_{1} \) homotopic paths on \( S \) . Then the lifts of \( {\gamma }_{0} \) and \( {\gamma }_{1} \) on \( W \) from the same initial point have the same terminal ... | Suppose, in particular, that the surface \( S \) is simply connected, i.e., that the fundamental group of \( S \) is trivial. In this case the monodromy theorem yields an interesting corollary:\n\nIf \( \left( {W, f}\right) \) is an unlimited covering surface of a simply connected surface \( S \) , then the mapping \( ... | Yes |
Lemma 2.1. Let \( \left( {W, f}\right) \) be a covering surface of \( S \) . For every \( p \in W \), there are parameter discs \( U \ni p \) and \( f\left( U\right) \), with local parameters \( k \) and \( h \) normalized by \( k\left( p\right) = h\left( {f\left( p\right) }\right) = 0 \), such that in \( U \), \[ h \c... | The proof is given in Ahlfors-Sario [1], p. 40. Conversely, if \( f : W \rightarrow S \) is a continuous mapping and the above condition holds, we conclude immediately that \( \left( {W, f}\right) \) is a covering surface of \( S \) . Thus this condition characterizes covering surfaces. | No |
Theorem 2.2. If the projection mapping \( f : W \rightarrow S \) is surjective and the covering group \( G \) of \( W \) over \( S \) is transitive, then \( W/G \) and \( S \) are homeomorphic. | Proof. We write \( \left\lbrack p\right\rbrack \in W/G \) for the equivalence class containing the point \( p \in W \) and prove that\n\n\[ \left\lbrack p\right\rbrack \rightarrow f\left( p\right) \]\n\n(2.2)\n\nis a homeomorphism of \( W/G \) onto \( S \) . First, it follows from \( f = f \circ g, g \in G \) , that (2... | Yes |
Theorem 2.3. The covering group of a universal covering surface \( W \) over a surface \( S \) is transitive. | Proof. Suppose that \( a \) and \( {a}^{\prime } \) are points of \( W \) which lie over the same point of \( S \) . Choose a point \( p \in W \), join \( a \) to \( p \) by a path on \( W \), project this path onto \( S \), and lift the projection back, but from the point \( {a}^{\prime } \) . Let \( {p}^{\prime } \) ... | Yes |
Theorem 2.4. The covering group of a universal covering surface of \( S \) is isomorphic to the fundamental group of \( S \) . | Proof. Given a point \( a \in W \), let \( \gamma \) be a closed path on \( S \) from \( f\left( a\right) \), and \( b \in W \) the terminal point of the lift of \( \gamma \) from \( a \) . Then \( a \) and \( b \) both lie over \( f\left( a\right) \) . By Theorem 2.3, there is a unique cover transformation \( {g}_{\ga... | Yes |
Theorem 2.6. Let \( W \) be a surface, \( G \) a properly discontinuous fixed point free group of homeomorphisms of \( W \) onto itself, and \( f : W \rightarrow W/G \) the canonical projection. Then\n\n1. \( W/G \) is a surface,\n\n2. \( \left( {W, f}\right) \) is an unlimited covering surface of \( W/G \) ,\n\n3. \( ... | Proof. By definition, \( f \) is continuous. If \( A \subset W \), then \( {f}^{-1}\left( {f\left( A\right) }\right) = \cup g\left( A\right) \) , \( g \in G \), from which we conclude that \( f \) is open.\n\nIn order to prove that \( W/G \) is a Hausdorff space we consider two different points \( f\left( a\right) \) a... | Yes |
Theorem 3.1. Let \( S \) be a Riemann surface and \( \left( {W, f}\right) \) a smooth covering surface of \( S \) . Then \( W \) carries a unique conformal structure which makes the projection mapping fanalytic. | Proof. Let \( H \) be the conformal structure of \( S \) . For every point \( p \in W \) we choose a neighborhood \( U \) of \( p \) such that \( f \mid U \) is injective and \( f\left( U\right) \) is contained in the domain of some \( h \in H \) . Then the atlas \( \{ h \circ \left( {f \mid U}\right) \mid p \in W\} \)... | Yes |
Theorem 3.2. Let \( W \) be a Riemann surface, \( G \) a properly discontinuous fixed point free group of conformal self-mappings of \( W \), and \( f : W \rightarrow W/G \) the canonical projection. Then the surface \( W/G \) carries a unique conformal structure which lifts to the original conformal structure of \( W ... | This follows immediately from the way the local parameters of \( W/G \) were defined in the proof of Theorem 2.6. In the situation of Theorem 3.2, the conformal structure of \( W \) is said to have been projected to \( W/G \) . If \( W \) is a given Riemann surface, we always regard the quotient \( W/G \) as the Rieman... | Yes |
Theorem 3.4. Given an arbitrary Riemann surface \( S \), let \( D \) be its universal covering surface, and \( G \) the covering group of \( D \) over \( S \). Then \( S \) is conformally equivalent to the Riemann surface \( D/G \). | Proof. It follows from Theorems 2.3,2.5, and 3.2 that the quotient \( D/G \) is a Riemann surface with the projected conformal structure. By Theorem 2.2, the mapping (2.2) is a homeomorphism of \( D/G \) onto \( S \). It is conformal, because the conformal structure of \( S \) is also obtained by projection from \( D \... | Yes |
Theorem 3.5. Two homeomorphisms \( {\varphi }_{i} : {S}_{1} \rightarrow {S}_{2}, i = 0,1 \), induce the same group isomorphisms if and only if they are homotopic. | Proof. Assume first that \( {\varphi }_{0} \) is homotopic to \( {\varphi }_{1} \) . Let \( h \) be a homotopy from \( {\varphi }_{0} \) to \( {\varphi }_{1} \) and \( {f}_{t} \) a lift of \( h\left( {., t}\right) ,0 \leq t \leq 1 \), such that \( {f}_{t} \) is a homotopy between \( {f}_{0} \) and \( {f}_{1} \) .\n\nCh... | Yes |
Lemma 4.1. For a Kleinian group \( G \), every point \( \zeta \in L \) is the cluster point of each orbit \( G\left( z\right) \), with the possible exception of \( z = \zeta \) and one other point \( z \in L \) . | We first deduce from this lemma that if \( G \) is not elementary, every point of \( L \) is the cluster point of other limit points. Hence, \( L \) is then always a perfect set. It follows that for Möbius groups there is a striking dichotomy: Either the limit set contains at most two points or else it contains uncount... | No |
Lemma 4.2. Let \( G \) be a Kleinian group such that \( \Omega \) has an invariant component \( A \) which is a Jordan domain different from a disc. Then \( \partial A \) does not have a tangent at a fixed point of a loxodromic element of \( G \) . | Proof. Assume that the tangent exists at a fixed point of a loxodromic element \( g \in G \) . We may suppose without loss of generality that the fixed point of \( g \) lies at \( z = 0 \), that the tangent at \( z = 0 \) is the real axis and that \( \infty \) is the repulsive fixed point of \( g \) . Then \( g\left( z... | Yes |
Theorem 4.2. The boundary of an invariant component of a quasi-Fuchsian group is either a circle or a Jordan curve which fails to have a tangent on an everywhere dense set. | Proof. First, if \( A \) denotes an invariant component, we clearly have \( \partial A \subset \partial \Omega \) . From (4.3) we then conclude that \( \partial A \subset L \) . If the group is not Fuchsian, it always contains loxodromic elements (Lehner [1], p. 107). By (4.4), we have in this case \( \partial A \subse... | No |
Theorem 5.1. Let \( S \) be a Riemann surface and \( G \) the covering group of the upper half-plane \( H \) over \( S \) . Then \( S \) is compact if and only if the Dirichlet regions of \( G \) are bounded in the hyperbolic metric of \( H \) . | Proof. Suppose first that \( S \) is compact. Let \( N \) be a Dirichlet region with center \( a \) . We consider the hyperbolic discs \( {D}_{n} = \{ z \mid h\left( {z, a}\right) < n\}, n = 1,2,\ldots \) Their projections on \( S = H/G \) form an open covering of \( S \) . Since \( S \) is compact, there is an \( n \)... | Yes |
Theorem 5.2. The covering group of the upper half-plane over a compact Riemann surface is finitely generated and of the first kind. | Proof. Let \( S \) be a compact Riemann surface and \( G \) the covering group of \( H \) over \( S \) . The vertices of a Dirichlet region of \( G \) cannot have a limit point in \( H \) . Hence, by Theorem 5.1, a Dirichlet region for \( G \) has a finite number of sides. We conclude using Theorem 4.1 that \( G \) is ... | Yes |
Theorem 5.4 (Riemann-Roch Theorem). On a compact Riemann surface of genus \( p \), every divisor \( D \) satisfies the equation\n\n\[ \n\dim D = \dim \left( {-D - {D}_{1}}\right) - \deg D - p + 1.\n\] | Let us first apply (5.6) for \( D = - {D}_{1} \) . Then, by (5.5) and (5.2), \( p = 1 + \) \( \deg {D}_{1} - p + 1 \), so that \( \deg {D}_{1} = {2p} - 2 \) . By our previous remark, we have\n\n\[ \n\deg {D}_{{\varphi }_{1}} = {2p} - 2\n\]\n\nfor every meromorphic \( \left( {1,0}\right) \) -differential \( {\varphi }_{... | No |
Theorem 5.5. On a compact Riemann surface of genus \( p \), the space of holomorphic quadratic differentials has dimension 1 if \( p = 1 \) and \( {3p} - 3 \) if \( p > 1 \) . | Proof. In the case \( p = 1 \), the Riemann-Roch theorem is not needed to determine the dimension of \( Q \) . We saw in 4.1 that cover transformations are translations \( z \rightarrow z + m{\omega }_{1} + n{\omega }_{2}, m, n \in \mathbb{Z} \) . Formula (3.5) shows, therefore, that \( \varphi \) is a holomorphic quad... | Yes |
Theorem 7.1. Every point of a Riemann surface has a neighborhood in which any two points can be joined by a unique shortest curve. | Proof. Let a point \( p \in S \) be given and suppose first that \( p \) is regular. Let \( V \) be the maximal disc around \( p \) and \( \{ w\left| \right| w \mid < r\} \) its image under a natural parameter \( w = \Phi \left( z\right) \) . Let \( {V}_{0} \subset V \) be the preimage of \( \left| w\right| < r/2 \), a... | Yes |
Lemma 7.1 (Teichmüller’s Lemma). Let \( \varphi \) be holomorphic in the closure of a domain \( A \) in the complex plane which is bounded by a simple closed polygon in the \( \varphi \) -metric, whose sides \( {\gamma }_{j} \) form the angles \( {\theta }_{j} \) at the vertices. If \( {m}_{i} \) and \( {n}_{j} \) deno... | Proof. On \( {\gamma }_{j} \) we have \( \arg \left( {\varphi \left( z\right) d{z}^{2}}\right) = \) constant, and so\n\n\[ d\arg \varphi \left( z\right) + {2d}\left( {\arg {dz}}\right) = 0. \]\n\nThe argument of the tangent vector \( {dz} \) increases by \( {2\pi } - \sum \left( {\pi - {\theta }_{j}}\right) \) after a ... | Yes |
Lemma 7.2. Let \( S = G \) be a simply connected domain in the complex plane and \( {z}_{1} \) and \( {z}_{2} \) points of \( G \) . Then there exists at most one geodesic from \( {z}_{1} \) to \( {z}_{2} \) . | Proof. Let us assume that there are two geodesics joining \( {z}_{1} \) and \( {z}_{2} \) in, \( G \) . If they do not coincide we can find two subarcs, both from a point \( a \) to a point \( b \), which form a simple closed polygon. The angle condition (7.3) is satisfied at the vertices, except possibly at the two po... | Yes |
Lemma 7.3. In a simply connected subdomain of the complex plane every maximal geodesic is a cross-cut. | Proof. Let \( \gamma \) be a maximal geodesic in a simply connected plane domain \( S = G \) . Fix a point \( {z}_{0} \in \gamma \) and represent a ray of \( \gamma \) with the initial point \( {z}_{0} \) by using its arclength \( u \) as parameter, \( 0 \leq u < {u}_{\infty } \) . Assume that \( \gamma \left( u\right)... | Yes |
Theorem 7.3. Let \( S \) be a compact Riemann surface and \( p \) and \( q \) points of \( S \). Then each homotopy class of curves joining \( p \) and \( q \) on \( S \) contains a unique shortest (hence geodesic) arc. | Proof. As in the proof of Theorem 7.2, we may replace \( S \) by its universal covering surface \( D \). Let two points \( {z}_{1} \) and \( {z}_{2} \) of \( D \) be given. Since the distance from \( {z}_{1} \) and \( {z}_{2} \) to \( \partial D \) is infinite, we can find a Jordan domain \( G,\bar{G} \subset D \), suc... | Yes |
Lemma 7.4. Let \( S \) be a compact Riemann surface, \( f : S \rightarrow S \) a homeomorphism homotopic to the identity, and \( \alpha \) a horizontal arc. Then there is a constant \( M \) , which does not depend on \( \alpha \), such that\n\n\[ l\left( {f\left( \alpha \right) }\right) \geq l\left( \alpha \right) - {2... | Proof. Let \( h : S \times \left\lbrack {0,1}\right\rbrack \rightarrow S \) be a homotopy from the identity mapping to \( f \) . Fix a point \( p \in S \) and denote by \( {\widetilde{\gamma }}_{p} \) the path \( t \rightarrow h\left( {p, t}\right) \) . Let \( {\gamma }_{p} \) be the (unique) geodesic in the homotopy c... | Yes |
Theorem 1.1. Let \( \mu \) be a Beltrami differential on a Riemann surface \( S \) . Then there is a quasiconformal mapping of \( S \) onto another Riemann surface with complex dilatation \( \mu \) . The mapping is uniquely determined up to a conformal mapping. | Proof. We consider \( \mu \) as a Beltrami differential for the covering group \( G \) of \( D \) over \( S \) . By Theorem I.4.4, there is a quasiconformal mapping \( f : D \rightarrow D \) with complex dilatation \( \mu \) . Since (1.1) holds, \( f \) and \( f \circ g \) have the same complex dilatation for every \( ... | Yes |
Theorem 1.2. Let \( S \) and \( {S}^{\prime } \) be Riemann surfaces with non-elementary covering groups \( G \) and \( {G}^{\prime },{\varphi }_{i} : S \rightarrow {S}^{\prime }, i = 0,1 \), two quasiconformal mappings, and \( {f}_{0} \) a lift of \( {\varphi }_{0} \) . Then \( {\varphi }_{0} \) and \( {\varphi }_{1} ... | Proof. Suppose first that there is a lift \( {f}_{1} \) of \( {\varphi }_{1} \) such that \( {f}_{1} = {f}_{0} \) on the limit set \( L \) of \( G \) . Because \( {f}_{0} \) and \( {f}_{1} \) map \( L \) onto the limit set \( {L}^{\prime } \) of \( {G}^{\prime } \) and because \( L \) is invariant under \( G \), we the... | Yes |
Theorem 1.3. Let \( S \) be a Riemann surface with a non-elementary covering group. If \( f : S \rightarrow S \) is a conformal mapping homotopic to the identity, then \( f \) is the identity mapping. | Proof. By Theorem IV.3.5, \( f \) and the identity mapping of \( S \) induce the same group isomorphism of the covering group of \( D \) over \( S \) . By Theorem 1.2, \( f \) has a lift which is the identity mapping of \( D \) . Hence, the projection \( f \) itself is the identity mapping. | Yes |
Theorem 1.4. Two quasiconformal mappings \( {\varphi }_{i} : S \rightarrow {S}^{\prime }, i = 0,1 \), are homotopic modulo the boundary if and only if they can be lifted to mappings of \( D \) which agree on the boundary. | Proof. Assume first that \( {\varphi }_{0} \) and \( {\varphi }_{1} \) are homotopic modulo the boundary. If \( {f}_{0} \) is a lift of \( {\varphi }_{0} \), then the lift \( {f}_{1} \) of \( {\varphi }_{1} \) homotopic to \( {f}_{0} \) through the lifted homotopy agrees with \( {f}_{0} \) on the set \( B \) . The mapp... | Yes |
Theorem 1.5. Let \( S \) and \( {S}^{\prime } \) be compact, topologically equivalent Riemann surfaces. Then every homotopy class of sense-preserving homeomorphisms of \( S \) onto \( {S}^{\prime } \) contains a quasiconformal mapping. | Proof. Let \( f : S \rightarrow {S}^{\prime } \) be a sense-preserving homeomorphism. Since \( S \) is compact, it has a finite covering by domains \( {U}_{1},{U}_{2},\ldots ,{U}_{n} \), such that \( {U}_{k} \) is conformally equivalent to the unit disc and \( \partial {U}_{k} \) is an analytic curve. Set \( {f}_{0} = ... | Yes |
Theorem 2.1. Let \( {f}_{0} : S \rightarrow {S}^{\prime } \) be a quasiconformal mapping and \( F \) the class of all quasiconformal mappings of \( S \) onto \( {S}^{\prime } \) homotopic to \( {f}_{0} \) . Then \( F \) contains an extremal mapping, i.e., one with smallest maximal dilatation. | Proof. Let \( D \) be a universal covering surface of \( S \) . The theorem is trivial if \( D \) is the extended plane or if \( D \) is the complex plane and \( S \) is non-compact. In the case where \( D \) is the complex plane and \( S \) is compact, the theorem will be proved in 6.4. Hence, we may assume that \( D ... | No |
Theorem 2.2. The Teichmüller space \( {T}_{S} \) is pathwise connected. | Proof. The geodesic \( t \rightarrow \left\lbrack {\mu }_{t}\right\rbrack \) is a path joining the origin to the point \( p \) in \( {T}_{s} \) ; the path \( t \rightarrow \left\lbrack {t\mu }\right\rbrack \) of \( {T}_{S} \) also has this property. | No |
Theorem 2.3. The conformal structures \( {H}_{1} \) and \( {H}_{2} \) induced by the Beltrami differentials \( {\mu }_{1} \) and \( {\mu }_{2} \) on the Riemann surface \( S \) are deformation equivalent if and only if \( {\mu }_{1} \) and \( {\mu }_{2} \) determine the same point in the Teichmüller space \( {T}_{S} \)... | Proof. Let \( {f}_{i}, i = 1,2 \), be quasiconformal mappings of \( S \) with complex dilatations \( {\mu }_{i} \) . If \( \varphi : \left( {S,{H}_{1}}\right) \rightarrow \left( {S,{H}_{2}}\right) \) is a conformal mapping homotopic to the identity, we first conclude that the mapping\n\n\[ h = {f}_{2} \circ \varphi \ci... | Yes |
Theorem 2.4. On a compact Riemann surface \( S \), every conformal structure is deformation equivalent to a structure induced by a Beltrami differential of S. | Proof. Let \( H \) be the given and \( {H}^{\prime } \) an arbitrary conformal structure on \( S \) . By Theorem 1.5, there is a quasiconformal mapping \( f : \left( {S, H}\right) \rightarrow \left( {S,{H}^{\prime }}\right) \) which is homotopic to the identity. Let \( f \) have the complex dilatation \( \mu \) . Then ... | Yes |
Theorem 2.5. The Teichmüller space of a compact Riemann surface is isomorphic to the set of equivalence classes of conformal structures modulo deformation. | This result can also be expressed in somewhat different terms. Let \( \mathcal{H}\left( S\right) \) denote the set of all conformal structures of \( S \) . The group Homeo \( {}^{ + }\left( S\right) \) consisting of all sense-preserving homeomorphic self-mappings of \( S \) acts on \( \mathcal{H}\left( S\right) \) : If... | Yes |
Theorem 2.6. The Teichmüller spaces of two quasiconformally equivalent Riemann surfaces are isometrically bijective. | Proof. Let \( S \) and \( {S}^{\prime } \) be Riemann surfaces and \( h \) a quasiconformal mapping of \( S \) onto \( {S}^{\prime } \) . The mapping \( f \rightarrow f \circ {h}^{-1} \) is a bijection of the family of all quasiconformal mappings \( f \) of \( S \) onto the family of all quasiconformal mappings of \( {... | Yes |
Theorem 2.7. The Riemann space is the quotient of the Teichmüller space by the modular group. | Proof. Assume first that the points \( \left\lbrack f\right\rbrack \) and \( \left\lbrack g\right\rbrack \) of \( {T}_{S} \) are equivalent under \( \operatorname{Mod}\left( S\right) \) . We then have a quasiconformal mapping \( h : S \rightarrow S \) such that \( f \circ {h}^{-1} \) is equivalent to \( g \) . But this... | Yes |
Theorem 3.1. The Beltrami differentials \( \mu \) and \( v \) of \( S \) are equivalent if and only if \( {f}^{\mu }\left| {\mathbb{R} = {f}^{v}}\right| \mathbb{R} \) or if and only if \( {f}_{\mu }\left| {H = {f}_{v}}\right| H \) . | Proof. Let us first assume that \( \mu \) and \( v \) are equivalent. Let \( \varphi \) and \( \psi \) be quasiconformal mappings of \( S \) which lift to \( {f}^{\mu } \) and \( {f}^{v} \), respectively. Then there is a conformal map \( \eta : \varphi \left( S\right) \rightarrow \psi \left( S\right) \) such that \( \e... | Yes |
Lemma 3.1. Let \( \left\lbrack {\mu }_{n}\right\rbrack \rightarrow \left\lbrack \mu \right\rbrack \) in \( {T}_{S},{\begin{Vmatrix}{\mu }_{n}\end{Vmatrix}}_{\infty } \leq k < 1 \), and \( {\mu }_{n} \rightarrow v \) a.e. Then \( \left\lbrack \mu \right\rbrack = \) \( \left\lbrack v\right\rbrack \) in \( {T}_{S} \) . | Proof. Let \( {\lambda }_{n} \in \left\lbrack {\mu }_{n}\right\rbrack \) be an extremal complex dilatation for which \( {\tau }_{S}\left( \left\lbrack {\mu }_{n}\right\rbrack \right. \) , \( \left. \left\lbrack \mu \right\rbrack \right) = \operatorname{artanh}{\begin{Vmatrix}\left( {\lambda }_{n} - \mu \right) /\left( ... | Yes |
Theorem 3.3. The mapping \( g \rightarrow {f}_{\mu } \circ g \circ {f}_{\mu }^{-1} \) defines an isomorphism of the covering group \( G \) onto a group \( {G}_{\mu } \) of Möbius transformations acting on the quasidisc \( {f}_{\mu }\left( {H}^{\prime }\right) \) . | Proof. Consider the quasiconformal mapping \( {f}_{\mu } \circ g \circ {f}_{\mu }^{-1}, g \in G \), of the plane. It is conformal in \( {f}_{\mu }\left( H\right) \), because \( {f}_{\mu } \mid H \) is conformal. Since \( \mu \) is a Beltrami differential for \( G \), the mappings \( {f}_{\mu } \) and \( {f}_{\mu } \cir... | Yes |
Theorem 3.5. The mapping\n\n\\[ \n\\left\\lbrack \\mu \\right\\rbrack \\rightarrow {f}^{\\mu } \\mid \\mathbb{R} \n\\] \n\n(3.4)\n\nis a homeomorphism of \\( \\left( {{T}_{S},{\\tau }_{S}}\\right) \\) onto \\( \\left( {X\\left( G\\right) ,\\rho }\\right) \\) . | Proof. By Theorem 3.1, the mapping (3.4) is well defined and injective. By Theorem 3.4, it is surjective.\n\nBy Theorem III.3.1, the mapping (3.4) is a homeomorphism of \\( \\left( {T,\\tau }\\right) \\) onto \\( \\left( {X,\\rho }\\right) \\) . Hence (3.4), which maps \\( {T}_{S} \\) bijectively onto \\( X\\left( G\\r... | No |
Lemma 4.1. The following three conditions are equivalent:\n\n\\( {1}^{ \\circ }{S}_{{f}_{\\mu } \\mid H} \\) is a quadratic differential for \\( G \\) ;\n\n\\( {2}^{ \\circ }{f}_{\\mu } \\circ g \\circ {f}_{\\mu }^{-1} \\) agrees with a Möbius transformation in \\( {f}_{\\mu }\\left( H\\right) \\) for \\( g \\in G \\) ... | Proof. As we already remarked, the equivalence of \\( {1}^{ \\circ } \\) and \\( {2}^{ \\circ } \\) follows directly from (4.1). If \\( {2}^{ \\circ } \\) holds, i.e., if \\( {f}_{\\mu } \\circ g \\circ {f}_{\\mu }^{-1} = w \\) in \\( {f}_{\\mu }\\left( H\\right) \\), where \\( w \\) is a Möbius transformation, then \\... | Yes |
Theorem 4.1. The Teichmüller spaces satisfy the relation\n\n\[ T\\left( G\\right) = Q\\left( G\\right) \\cap T\\left( 1\\right) \] | Proof. The inclusion \( T\\left( G\\right) \\subset Q\\left( G\\right) \\cap T\\left( 1\\right) \) follows directly from the definitions. We choose an arbitrary point \( {S}_{f} \\in Q\\left( G\\right) \\cap T\\left( 1\\right) \) and prove that \( {S}_{f} \\in T\\left( G\\right) \) .\n\nLet \( w \) be a conformal mappi... | Yes |
Theorem 4.2. The set \( T\left( G\right) \) is closed in \( T\left( 1\right) \) . | Proof. The relation (4.4) is equivalent to \( T\left( G\right) = U\left( G\right) \cap T\left( 1\right) \) . Since \( U\left( G\right) \) is closed in \( Q\left( 1\right) \), the theorem follows. | Yes |
Theorem 4.3. The set \( T\left( G\right) \) is open in \( Q\left( G\right) \) . | Proof. This can be read from (4.4), since \( T\left( 1\right) \) is open in \( Q\left( 1\right) \) . | No |
Theorem 4.4. The ball\n\n\[ B\left( {0,2}\right) = \{ \varphi \in Q\left( G\right) \mid \parallel \varphi \parallel < 2\} \]\n\nlies in \( T\left( G\right) \) . | Proof. In III.4.3 we remarked that \( \{ \varphi \in Q\left( 1\right) \mid \parallel \varphi \parallel < 2\} \) lies in \( T\left( 1\right) \) (Theorem II.5.1). Hence, the theorem follows immediately from (4.4). | No |
Theorem 4.5. Every point of the Teichmüller space \( {T}_{S} \) can be represented by a real analytic Beltrami differential and by a real analytic quasiconformal mapping. | Proof. Let a point \( \left\lbrack \mu \right\rbrack = p \in {T}_{S} \) be given. Suppose first that \( p \) can be represented by a quasiconformal mapping whose maximal dilatation is \( < 2 \) . Then \( p \) lies in the set (4.7), and so \( p \) can be represented by \( z \rightarrow - 2{y}^{2}\varphi \left( \bar{z}\r... | Yes |
For every \( {s}_{\mu } \in T\left( G\right) \), the ball\n\n\[ B\left( {{s}_{\mu },{\sigma }_{I}\left( {A}_{\mu }\right) }\right) = \left\{ {\varphi \in Q\left( G\right) \mid q\left( {{s}_{\mu },\varphi }\right) < {\sigma }_{I}\left( {A}_{\mu }\right) }\right\} \]\n\nis contained in \( T\left( G\right) \) . | In III.5.3 we proved that \( \left\{ {\varphi \in Q\left( 1\right) \mid q\left( {{s}_{\mu },\varphi }\right) < {\sigma }_{I}\left( {A}_{\mu }\right) }\right\} \) lies in \( T\left( 1\right) \) . Consequently, the theorem follows immediately from (4.4). | No |
Theorem 4.8. The mapping\n\n\[ \left\lbrack \mu \right\rbrack \rightarrow {S}_{{f}_{\mu } \mid H} \]\n\n(4.15)\n\nis a homeomorphism of \( \left( {{T}_{S},{\tau }_{S}}\right) \) onto \( \left( {T\left( G\right), q}\right) \) . | Proof. By Theorem III.4.1, this mapping is a homeomorphism of \( \left( {{T}_{S},\tau \mid {T}_{S}}\right) \) onto \( T\left( G\right) \) . By Theorem 4.7, the metrics \( {\tau }_{S} \) and \( \tau \mid {T}_{S} \) are equivalent, and the theorem follows. | Yes |
Theorem 5.1. The function\n\n\\[ \n\mu \rightarrow \Lambda \left( \mu \right) = {S}_{{f}_{\mu } \mid H} \n\\]\n\n(5.1)\n\nwhich maps the open unit ball \\( B\\left( G\\right) \\) of the space of measurable \\( \\left( {-1,1}\\right) \\) - differentials for \\( G \\) into the space \\( Q\\left( G\\right) \\) of holomorp... | Proof. We have already seen that \\( Q\\left( G\\right) \\) is a Banach space. The ball \\( B\\left( G\\right) \\) is an open subset of the Banach space \\( {L}^{\\infty }\\left( G\\right) \\) of measurable \\( \\left( {-1,1}\\right) \\) -differentials for \\( G \\) with finite \\( {L}^{\\infty } \\) -norm. Fix \\( \\m... | Yes |
Theorem 5.2. The atlas\n\n\\[ \n\\left\\{ {\\left( {{V}_{\\mu },{h}_{\\mu }}\\right) \\mid \\mu \\in B\\left( G\\right) }\\right\\} \n\\]\n\n(5.8)\n\ndefines a complex analytic structure on the Teichmüller space \\( {T}_{S} \\) . The Bers imbedding \\( \\left\\lbrack \\mu \\right\\rbrack \\rightarrow {\\left. {S}_{{f}_... | Proof. Assuming that \\( {V}_{\\mu } \\) and \\( {h}_{\\mu } \\) are defined by (5.6) and (5.7), we choose two elements \\( {\\mu }_{1} \\) and \\( {\\mu }_{2} \\) of \\( B\\left( G\\right) \\) such that \\( {V}_{{\\mu }_{1}} \\cap {V}_{{\\mu }_{2}} \\) is not empty. In \\( {h}_{{\\mu }_{1}}\\left( {{V}_{{\\mu }_{1}} \... | Yes |
Theorem 5.3. The canonical projection\n\n\\[ \n\\pi : B\\left( G\\right) \\rightarrow {T}_{S}\n\\] \nis holomorphic, and it has local holomorphic sections everywhere in \\( {T}_{s} \\) . | Proof. First of all, we have\n\n\\[ \n{h}_{\\mu } \\circ \\pi = {\\Lambda }_{\\mu } \\circ {\\widetilde{\\alpha }}_{\\mu }\n\\] \n\nSince \\( {\\Lambda }_{\\mu } \\) and \\( {\\widetilde{\\alpha }}_{\\mu } \\) are holomorphic, it follows that \\( \\pi \\) is holomorphic.\n\nNext, let us consider the mapping\n\n\\[ \n{\... | Yes |
Theorem 5.4. The Bers imbedding \( \lambda : {T}_{S} \rightarrow T\left( G\right) \) is biholomorphic. | Proof. Suppose first that \( Q\left( G\right) \) is finite dimensional. (By IV.5.5, this is the case if \( S \) is compact; cf. also 9.7.) We then conclude directly from Theorem 5.2 that \( \lambda \) is biholomorphic using the theorem by which a holomorphic bijection is always biholomorphic in finite dimensional manif... | Yes |
Theorem 5.5. Quasiconformally equivalent Riemann surfaces have isometrically and biholomorphically isomorphic Teichmüller spaces. | Proof. Let \( S = {H}^{\prime }/G \) and \( {S}^{\prime } = {H}^{\prime }/{G}^{\prime } \) be quasiconformally equivalent Riemann surfaces. We consider a lift of a quasiconformal mapping of \( S \) onto \( {S}^{\prime } \) to a self-mapping of the lower half-plane. There is no loss of generality in assuming that the li... | No |
Theorem 5.6. The elements of the modular group \( \operatorname{Mod}\left( S\right) \) are biholomorphic automorphisms of the Teichmüller space \( {T}_{S} \) . | Proof. By Theorem 5.5, a quasiconformal mapping between the Riemann surfaces \( S \) and \( {S}^{\prime } \) induces a biholomorphic isomorphism \( {T}_{S} \rightarrow {T}_{{S}^{\prime }} \) . The elements of the modular group are such isomorphisms induced by quasi-conformal self-mappings of \( S \) . | Yes |
Lemma 6.1. Let \( \left( {{\omega }_{1},{\omega }_{2}}\right) \) be a base of \( G \) . Then \( \left( {{\omega }_{1}^{\prime },{\omega }_{2}^{\prime }}\right) \) is a base of \( G \) if and only if\n\n\[{\omega }_{1}^{\prime } = a{\omega }_{1} + b{\omega }_{2},\;{\omega }_{2}^{\prime } = c{\omega }_{1} + d{\omega }_{2... | Proof. The validity of (6.1) with integral coefficients is clearly a necessary condition. It becomes sufficient if (6.1) can be solved with respect to \( {\omega }_{1} \) and \( {\omega }_{2} \) so that \( {\omega }_{1} \) and \( {\omega }_{2} \) are linear combinations of \( {\omega }_{1}^{\prime } \) and \( {\omega }... | Yes |
Lemma 6.2. Let \( S \) be a torus and \( p \) and \( q \) arbitrary points of \( S \) . Then there is a conformal mapping \( f : S \rightarrow S \) homotopic to the identity such that \( f\left( p\right) = q \) . | Proof. Let \( \pi : \mathbb{C} \rightarrow S = \mathbb{C}/G \) be the canonical projection and \( z \in {\pi }^{-1}\{ p\}, w \in \) \( {\pi }^{-1}\{ q\} \) . A translation commutes with every \( g \in G \) . Therefore, the mapping \( \zeta \rightarrow \zeta + t\left( {w - z}\right) \) can be projected to a conformal ma... | Yes |
Theorem 6.1. Let \( S = \mathbb{C}/G \) and \( {S}^{\prime } = \mathbb{C}/{G}^{\prime } \) be tori and \( \theta : G \rightarrow {G}^{\prime } \) an isomorphism. Then there is a homeomorphism of \( S \) onto \( {S}^{\prime } \) which induces \( \theta \) . | Proof. Let \( \left( {{\omega }_{1},{\omega }_{2}}\right) \) be a base of \( G \) and suppose that \( \left( {{\omega }_{1},{\omega }_{2}}\right) \rightarrow \left( {{\omega }_{1}^{\prime },{\omega }_{2}^{\prime }}\right) \) under \( \theta \) . Consider the affine transformation \( \alpha \) which fixes 0 and maps \( ... | Yes |
Theorem 6.2. Let \( S = \mathbb{C}/G \) and \( {S}^{\prime } = \mathbb{C}/{G}^{\prime } \) be tori and \( \left( {{\omega }_{1},{\omega }_{2}}\right) \) and \( \left( {{\omega }_{1}^{\prime },{\omega }_{2}^{\prime }}\right) \) normalized bases of \( G \) and \( {G}^{\prime } \) . Then \( S \) and \( {S}^{\prime } \) ar... | Proof. We just showed that \( S \) and \( {S}^{\prime } \) are conformally equivalent if and only if there is a \( \lambda \neq 0 \) such that \( \left( {\lambda {\omega }_{1},\lambda {\omega }_{2}}\right) \) is a base of \( {G}^{\prime } \) . From what we said at the end of 6.1 it follows that this is the case if and ... | Yes |
Lemma 6.3. Let \( \theta : G \rightarrow {G}^{\prime } \) be an isomorphism generated by a normalized \( K \) - quasiconformal mapping \( f \) . Then\n\n\[{\delta }_{\theta } \leq \frac{1}{2}\log K\]\n\nEquality holds if and only if \( f \) is the affine transformation generating \( \theta \) . | Proof. Let \( w \) be the affine normalized mapping which generates \( \theta \) . If \( w\left( z\right) = \) \( \lambda \left( {z + \mu \bar{z}}\right) \), we see from (6.5) that \( \left| \mu \right| = \left| {{\tau }^{\prime } - \tau }\right| /\left| {{\tau }^{\prime } - \bar{\tau }}\right| \), where \( \tau = {\om... | Yes |
Theorem 6.4. The mapping \( \psi : {T}_{S} \rightarrow H \), defined by\n\n\[ \psi \left( \left\lbrack \varphi \right\rbrack \right) = f\left( {\omega }_{1}\right) /f\left( {\omega }_{2}\right) \]\n\nwhere \( f \) is the normalized lift of \( \varphi \), is a bijective isometry of \( {T}_{S} \) onto the upper half-plan... | Proof. The mapping \( \psi \) is injective: If \( \psi \left( {p}_{1}\right) = \psi \left( {p}_{2}\right) \), there are mappings \( {\varphi }_{i} \in {p}_{i}, i = 1,2 \), whose normalized lifts \( {f}_{i} \) satisfy the equations \( {f}_{1}\left( {\omega }_{i}\right) = \lambda {f}_{2}\left( {\omega }_{i}\right) \) , \... | Yes |
Theorem 6.5. The mapping\n\n\[ \left\lbrack z\right\rbrack \rightarrow z \]\n\n(6.13)\n\nis a bijective isometry of the Teichmüller space \( {T}_{S} \) onto the hyperbolic unit disc D. | Proof. The theorem follows immediately from the fact that (6.12) is a bijective isometry of \( {T}_{S} \) onto the hyperbolic upper half-plane. | Yes |
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